Read the result in its research context.

Result

All four actual Pq at fixed S01 satisfy Pq=Z+|vq><vq| with common rank-one Z and overlap (11+4sqrt6)/25. A spectral polynomial of sum Pq constructs Z; products generate full M4 on Z-perp, hence mark algebra C+M4 and commutant C². All six pairs are noncommuting and admit no common sharp pointer refinement. Original D1 couples Z to its complement with squared crossnorms7882461/4398046511104 and3201175/13194139533312, giving full response algebra M5 and scalar commutant.

Required assumptions

• Given local E8_1 vacuum representation and explicit cocycle

  Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.

• Actual native G31 event representation

  Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.

• Symmetric quartic and actual erasure intertwiner

  Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.

• Actual positive Cartan quartic Fock embedding

  Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.

• Original displayed J-equivariant root isometry

  Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.

• Full existing local E8 field algebra with observer five-code

  Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.

• Intrinsic event geometry without marked clock as selector

  User159c5316 17Oct replaces automatic instrument/whitening-origin next step with intrinsic structure test. In accepted full unitaryE8_1 fieldnet use actual quarticproducts,eventrelations,*-operation,centralOPEnormalization. No originalpolar/P2,desired3+2,markedq,selectedclock orphysicalrealstate rule as construction criterion. Compare originalgeometry/clocks only afterwards. Transportmetrics withbasis changes; distinguish centralalgebraicOPE data andstate-dependentGNS responses. Only states with definedcommonfieldmoments; positivefiniteenergycontrols not automaticallyRP/clustering/finiterelativeactionP1seeds. Retain full localproducts,composition,correlations; no assumed independenttensorphysics.

• Original seam/state data on actual60 flags without fitted selection

  User17Oct: test originalseam/statedata onactual60 flags P(q,S); no fittedclock,coupling,newselectionrule. Additional EXISTING data topreviousunmarkedobject: fullchargedinternalv=(0^5,1/2^3),deck square,originalE8vacuum; optionaloriginalR_neut/retained48familyomission explicitgivenpremise,all4cases. SEPARATEactualoriginalfoldphasesalpha1/4,3/4withfixedexistingM,d=M((j+alpha)/8−1/2);disshift/timediagnostic,notautomaticallyfullCARcovarianceorE8vacuumexpectation. Fullroot/fieldaction,nocompressiononly. NopostulatedphysicalTheta,field/state/normalorderingdictionaryorP1selection.

• Fixed conformal current reference in original fold-state transport

  Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.

• Four starts at fixed state versus joint datum transport

  Comparison scope: q2,3,4,5 and globalS01. Retain originalFCαF*,M/O,fullE8fields,nativeJ andactualchargedfold,time,adjoint,normalordering,erasure. Separate passive whole-presentation changes, active mathematical field automorphisms and symmetries of FIXED given source state. Marks are actual intrinsic quartic rays/polar planes,not chosen couplings; equivalence retains stated field/coordinate/reference object. Do not postulate physical access orP1selection from comparison.

• Original conditional pointer template and supplied context

  Original text assumes A_obs subset A_rec and complete orthogonal pointers, then writes M_O(rho)=sum P_i rho P_i tensor |i><i|. Optional context Q supplies rho0, observables, time and readout. No identification with native Pq or physical P1 implementation is assumed.

Original evidence

• FOUR-POLAR-CONTEXT-ALGEBRA.txt lines 27–118

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/joint-native-algebra-selection-20261016/FOUR-POLAR-CONTEXT-ALGEBRA.txt

  f9bb309bae0d65b31df6a38899c526547f027b69eb89b34800497e72b4ea9bdb

• check_four_polar_context_algebra.py lines 1–94

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/joint-native-algebra-selection-20261016/check_four_polar_context_algebra.py

  7e30b0c2738d77b183f51cbb137480f07249a30f763d8b9d3acd824bb4b529cb

• four_polar_context_algebra_results.json lines 1–55

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/joint-native-algebra-selection-20261016/four_polar_context_algebra_results.json

  269ffa3c7124f93f36db1c437ab22046a84d2d9bca7531396755490b41924a92

• four_polar_context_algebra_review.json lines 1–55

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/joint-native-algebra-selection-20261016/four_polar_context_algebra_review.json

  5dd3fa191fa307440cbc4be5a670c43ad243a233b2526e287e388da3f1377f5d

finding:four-polar-context-algebra-and-source-crosscoupling